...Is there such a theory?

(If there is, I imagine it comes from the fact that Elamite was used as the Persian court language from time to time, but I think it's far more likely that Cyrus was Median if he wasn't Persian: Median was also probably a Persian court language, though it's unattested except in some Old Persian loanwords.)
Sure thing. This is very long. Warning ahead.
1. Cyrus primarily had inscriptions done in Akkadian and Elamite, the two administrative languages of Elam. In both, he refers to himself as the King of Anshan. This is a title the Elamite kings used to refer to themselves for centuries prior to Cyrus. In Elamite, they were the kings of Anshan and Susa and in Akkadian they were the kings of Susa and Anshan. Anshan got primacy in their own language and is believed to be their homeland before they moved into the more fertile lowland. In these incriptions Cyrus refers to himself as Cyrus, son of Cambyses (
Kambūjiya). In places, he goes back further saying Cyrus son of Cambyses, son of Cyrus, son of Teispes (
Cišpiš). Of Cyrus's Akkadian and Elamite inscriptions, they stop there in the genealogy. Put in a pin in this.
2. There are inscriptions in Pasargadae in Old Persian cuneiform. They read, "I am Cyrus, the King, an Achaemenid"; "Cyrus the Great King, son of Cambyses the King, an Achaemenid"; and "Cyrus the Great King, an Achaemenid." This seems straight forward in the sense that Elamites and Babylonians wouldn't care about Achaemenid, so he didn't bother putting that in his inscriptions. But the Behistun Inscription has a passage from Darius where he seems to claim to have invented Persian Cuneiform. "Says Darius the King. By the will of Ahuramazda, I made a script of a different sort, in Aryan, that previous did not exist." There's are alternative translations, that I'll get into when I rebut myself. If Darius invented the Persian script, that makes the inscriptions where Cyrus in his own words calls himself an Achaemenid
a forgery.
3. Darius came to power in suspicious circumstances. In the Behistun Inscription, he claims Bardiya, son of Cyrus the Great, was killed by his brother Cambyses, but no one knew. After Cambyses's death, a Magus named Gaumâta took Bardya's place, pretending to be him. Darius, by the grace of Ahuramazda, slew Gautama and became king. In doing so, he rightfully restored the kingdom to the Achaemenid line, as it was in days of old. Some who don't believe this story have hypothesized that this is imperial propaganda and that the other inscription is also a forgery. Together, they establish that Darius is a distant cousin of Cyrus and that the Achaemenid (Darius's ancestor) and his descendants rightfully possess the kingdom.
4. On religion, you say he's Zoroastrian. But he never invokes Ahuramazda, unlike Darius, who did it a lot. In the Cyrus Cylinder, he invokes the Babylonian god Mardok. Cyrus was also entombed. Under Zoroastrian beliefs, the dead are raised into a Tower of Silence, where they are given an "sky burial," i.e. left to be eaten by birds. Alternatively, you can place them in a raised stone tomb high above the ground. This is how Darius was buried.
5. Back to the names. Darius (
Dārayava(h)uš) has really obvious Iranian origins meaning "he who holds firm the good(ness)." His father, Hystaspes (
Vishtaspa) has an Iranian name that appears in the Avesta. His father, Arsames (
Aršāma) has an Old Persian name meaning hero. His father, Ariaramnes (
Ariyāramna) has an Iranian name that means "He who brings peace to the Aryans." His claimed father is Teispes, a common ancestor of Cyrus, according to Darius. Teispes (
Cišpiš) doesn't have an obvious Iranian root. It's thought it might come from a Mittani god (Theispas), but there's also a theory it comes from an Elamite name (
Za-iš-pi-iš-ši-ya). Achaemenes (
Haxāmaniš) is another very Iranian name, which means something like "having a friend's mind." On the other hand, Cyrus family does not have obvious Iranian names. Cyrus is
Kūruš. In Elamite, it was rendered Kurash. This is a bit odd because there were a lot of Elamite names that ended in "ush" and it wouldn't make any sense to render it as Kurash unless that's the original name. Also, there weren't any Persian names that ended in "ash," so it would make sense to change the Elamite name, but wouldn't make sense to change the Persian name. The Elamite name suggestion would be "one who bestows." Cambyses (
Kambūjiya) has been suggested to have an Elamite root (though, as I'll get to, it has a much stronger Iranian root). So, of the ancestors Darius claimed he had in his inscriptions, only one has a likely Persian name.
Does that seem convincing? Well, let me rebut.
1. There's no question that Cyrus's family had been in Anshan for a long time. Every source agrees with that. It's logical that he would adopt the naming convention of Anshan when writing in Elamite and the naming convention of Babylonia when writing in Akkadian. Moreover, in both places, it wasn't customary to trade back to a dynasty founder. You say your name and maybe go back a few generations. In the Babylonian chronicles, Cyrus is called the King of "Parsu." Anshan is in Parsa (it's about 46 miles from Pasargadae).
2. There are other widely accepted translations of Darius's inscription and they don't all say Darius invented the Old Persian script. Another possible translation of the inscription merely claims that he translated the specific
text into Aryan or even merely that he was proclaiming that the text was written in his own language. The last is the version on the Wikipedia page. That means that Cyrus's inscription might not be a forgery.
3. The Bardiya story is weird. But keep in mind that no ancient source accused Darius of lying. Also, there's not much need to invent Cyrus's Persian-ness. It would take a lot of risk in re-inventing the collective memory of people not far removed from Cyrus's life. Plus there's a least some evidence that Darius was part Elamite on his mother's side (though I don't have that particular source handy).
4. This one is pretty simple. When writing for Babylonians, you invoke Babylonian gods. As for Cyrus's burial, there's some evidence that Zoroastrian practice was not yet set. Besides, even if he wasn't Zoroastrian, that doesn't mean he wasn't Persian.
5. Names. Let's start with the easy one. Cambyses (Kambuziya) likely has the same root as the Sanskrit Kamboja. The Kamboja were an ancient tribe in India who likely came from Iran. (Kamboja is etymologically related to Cambodia, as that name comes from Sanskrit.) Teispes's name origin has been proposed as an Eastern Iranian root "chap" or "chishp," and meaning strength. Alternatively, he could have just had a Hurrian god name as they were an Indo-European people who may have interacted with ancient Iranians. Finally, there's Kurush the Great. There's a strong case that it means "youngest son" or "hero" (dating to traditions of the youngest son setting off to be a hero.) It could also be a tribal name--Indian texts later speak of a Kurus tribe in what is now Afghanistan. As for the ash/ush thing, there are two pretty simple explanations. The first is that there was an unrelated Elamite name of Kurash and Kurush was just translated into the already established, but unrelated name. The second is that he has an Elamite name because he was raised in an Elamite city.
There's something plausible in between. At some point the Achaemenid line became the rulers of Anshan. Cyrus, as king of Anshan, grew up in an Anshanite royal house and learned to do things in an Elamite way. The administrative language was Akkadian and the people spoke Persian, Elamite, and Akkadian. Darius was much more of an Aryan culturally (so closer to the Medes culturally than Cyrus). Thus when he came to power, he emphasized the Achaemenid roots and Aryan heritage. He used Persian and Aramaic as the languages of his empire and he ditched archaic Elamite terms. Cyrus used archaic (Elamite) names for Babylon (Tintir and Shuanna). Darius switched to calling it Babylon (Akkadian Bab-ilu). Cyrus used the archaic name "Gutium" to refer to Media and "Umman Manda" to refer to the Medians. Darius switched that to the Akkadian "Mandai" to refer to the Medes. By that point, Anshan wasn't as important and a Persian Empire that stretched from the Indus to the Hellespont needed to use standardized terms.
Source: Discovering Cyrus by Reza Zrghamee. It's in Appendix 1 and Appendix 2 with the accompanying endnotes. Some of the ideas come from Khodada Rezakhani in his (defunct) History of Iran Podcast.
And of course he's he's not Aryan. The Aryan's conquered India, and the evidence is certainly preponderant that Cyrus was a Zoroastrian from Iran (which of course doesn't rule out being Elamite--except he has an Iranian name

).
Aryan, in this case, is the same as Ancient/Classical Iranian. In Old Persian, it's Arya and in Avestan it's Airiia. The Aryans who (allegedly) conquered India are related to the Iranians. The Iranians referred to themselves as Aryan.